INTEGRATION OF GPS AND LEVELLING FOR SUBSIDENCE MONITORING STUDIES AT COSTA BOLIVAR OIL FIELDS, VENEZUELA J. LEAL October 1989 TECHNICAL REPORT NO. 144
Sep 27, 2015
INTEGRATION OF GPS ANDLEVELLING FOR SUBSIDENCE
MONITORING STUDIES ATCOSTA BOLIVAR OIL FIELDS,
VENEZUELA
J. LEAL
October 1989
TECHNICAL REPORT NO. 144
PREFACE
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INTEGRATION OF GPS AND LEVELLING FOR SUBSIDENCE MONITORING STUDIES
AT COSTA BOLIVAR OIL FIELDS, VENEZUELA
J. Leal
Department of Surveying Engineering University of New Brunswick
P.O. Box 4400 Fredericton, N.B.
Canada E3B 5A3
October 1989
J. Leal 1989
PREFACE
This technical report is a reproduction of a thesis submitted in partial fulfillment of the
requirements for the degree of Master of Science in Engineering in the Department of
Surveying Engineering, Apri11989. A few editorial modifications have been introduced by
Dr. Y.Q. Chen, and by Dr. Adam Chrzanowski who supervised the research. Funding was provided partially by Maraven S.A. of Venezuela, the Natural Sciences and
Engineering Research Council of Canada, and by the National Research Council of
Canada's Industrial Research Assistance Program agreement with Usher Canada Ltd.
As with any copyrighted material, permission to reprint or quote extensively from this
report must be received from the author. The citation to this work should appear as
follows:
Leal, J. (1989). Integration of Satellite Global Positioning System and Levelling for the Subsidence Monitoring Studies at the Costa Bolivar Oil Fields in Venezuela. M.Sc.E. thesis, Department of Surveying Engineering Technical Report No. 144, University of New Brunswick, Fredericton, New Brunswick, Canada, 124 pp.
ABSTRACT
Monitoring of ground subsidence has been traditionally performed by means of
geodetic levelling techniques. Geodetic levelling is slow and costly, requiring long
connection lines to stable areas, and higher densification in critical areas to properly depict
the deformation behaviour. The Global Positioning System (GPS) has been envisioned as an attractive alternative in the domain of deformation monitoring, bringing about potential
savings without significant deterioration in accuracy.
The Costa Bolivar oil fields in Venezuela have been subject to subsidence since 1926 at a rate of 20 em/year. The monitoring scheme has been based on geodetic levelling and an
already obsolete computational methodology. A full evaluation of the whole scheme has
revealed a total uncertainty of 20 to 30 nun at the 95% confidence level for the subsidence
determination and of 15 to 20 nun at the 95% confidence level for the absolute elevations.
A methodology to integrate GPS with levelling in order to modernize and optimize the
present monitoring scheme has been designed. The results of pilot tests to evaluate the real
accuracy of GPS in the area using WM101 receivers, show an accuracy of 29 mm
independent of baseline length. Accuracy standards developed for the optimal integration
reveal, however, that relative GPS accuracies in the order of 10 to 15 nun are needed for
compatible results with levelling. The results of an economic analysis on the designed
integration network shows savings in the order of 26% in the cost of one campaign which is
an indication of the feasibility of GPS when used in combination with levelling for
subsidence monitoring studies.
i i
TABLE OF CONTENTS
Page
ABSTRACT .................................................................... ii
TABLE OF CONTENTS iii
LIST OF FIGURES ....... ... . ............. ..... .. . . . ......... .. . . . . . ... .. . . . .. v
LIST OF TABLES . .. . . . . . .. . . . . ... .. . . . .. . ........... .............. .......... .. VI
ACKNOWLEDGEMENTS vii
1. INTRODUCTION ......................................................... 1
2. SUBSIDENCE DEFORMATION MODELLING ..................... 6
2.1 General Background
2.2 Deformation Models
6
8
2.3 Discrete Point Constant Velocity Model ......................... 13
2.4 Remarks on Other Approaches ................................... 16
3. EVALUATION OF THE PRESENT MONITORING SCHEME ... 18
3.1 Historic Synopsis
3.2 Network Description
3.3 Field Procedures
18
19
22
3.4 Data Processing Technique ....................................... 24 3.4.1 Computation of datum lines ............................... 24 3.4.2 Adjustment of main levelling network ................... 26 3.4.3 Computation of nodal lines ................................ 27 3.4.4 Computation of secondary lines .......................... 27 3.4.5 Additional remarks ......................................... 28
3.5 Economic Aspects ................................................. 28 3.5.1 Cost of levelling ........................................... 28 3.5.2 Cost of maintenance ....................................... 29 3.5.3 Cost of supervision and data processing ................ 29 3.5.4 Total estimated cost of one campaign .................... 30
3.6 Accuracy Evaluation ............................................... 30 3.6.1 Description of survey data ................................. 31 3.6.2 Accuracy of levelling surveys ............................. 31 3.6.3 Validity of the static network assumption ................ 37 3.6.4 Stability of the reference network ......................... 38 3.6.5 Final accuracy of elevations in single campaigns ........ 40
Page
3.6.6 Accuracy of the subsidence determination ................ 40 3.6.7 Final accuracy evaluation of the subsidence using the
UNB Generalized Method . . . . .. . . . . .. . . . . . .. . . . . . . . . . . . .. . . 45
4. ACCURACY OF GPS DERIVED HEIGHT DIFFERENCES ........ 49
4.1 Satellite Geometry
4.2 Observational Errors
49
50
4.2.1 Orbit related errors ................................... ........ 50 4.2.2 Tropospheric effect .......................................... 51 4.2.3 Ionospheric effect ............................................ 53 4.2.4 Carrier signal multipath and antenna imaging ............ 55 4.2.5 Antenna phase center variation ............................ 55 4.2.6 Other error sources .......................................... 56
4.3 Costa Bolivar GPS Campaigns .................................... 56 4. 3.1 Accuracy evaluation and summary of results ............. 57
4.4 Future Expectations 62
5. INTEGRATION OF GPS AND LEVELLING ......................... 65
5.1 Ellipsoidal Versus Orthometric Heights ........................... 65
5.2 The Problem of Gravity Variations ................................ 69
5.3 Integration Model 75
5.4 Design of Integrated Network and Field Surveys ............... 79
5.5 Accuracy Standards 82
5.6 Strategy for the Integration .......................................... 85 5.6.1 Field strategy ................. ................................. 85 5.6.2 Computational strategy ....................................... 87
5.7 Cost Analysis ......................................................... 88 5. 7.1 Initial investment cost in Maraven operations ............. 88 5. 7.2 Replacement of levelling by GPS .......................... 89
6. CONCLUSIONS AND RECOMMENDATIONS ...................... 94
7. REFERENCES .............................................................. 97
APPENDIX I (GPS and Levelling Data) ...................................... 104 APPENDIX II (Formula Derivation) APPENDIX III (Integration Example)
iv
108
112
LIST OF FIGURES
Page
Figure 1.1. Relative Location of the Costa Bolivar Oil Field [after Puig, 1984] 2
Figure 2.1. Examples of deformation with discontinuity 11
Figure 2.2. Subsidence Basins (Cumulative Subsidence 1926-1986) 12
Figure 3.1. Main Levelling Net 20
Figure 3.2. Detail showing Secondary and Nodal Lines 21
Figure 3.3. Computational Sequence Flow Chart 25
Figure 3.4. Historic Plot of Critical Bench Mark A (Lagunillas basin) 34
Figure 3.5. Historic Plot of Critical Bench Mark 608 (Bachaquero basin) 35
Figure 3.6. Historic Plot of Critical Bench Mark 215 (Lagunillas basin) 36
Figure 4.1. GPS Misclosures Campaign No. 1 58
Figure 4.2. GPS Misclosures Campaign No. 2 59
Figure 4.3. GPS Misclosures Campaign No.3 60
Figure 5.1. Local Geoid (Levelling - GPS) at the Costa Bolivar Network 68
Figure 5.2. Geometry of the Gravitational Attraction of a Body B Upon A 72
Figure 5.3. Designed Integration Network 81
Figure ILl. Conical Model 110
Figure III.l. Sample Levelling Network 113
v
LIST OFT ABLES Page
Table 3.1. Summary of Costa Bolivar Network Statistics 23
Table 3.2. Elevations of Reference Bench Marks [m] 26
Table 3.3. Statistical Summary of Survey Data 31
Table 3.4. Levelling Standard Deviations as Determined by MINQE 33
Table 3.5. Comparison of Subsidence Determinations 38
Table 3.6. "Best" Weighted Displacements for Reference BenchMarks 39
Table 3.7. Comparison of Elevations 41
Table 3.8. Summary of Standard Deviations for the Author Elevations 42
Table 3.9. Comparison of Subsidence Results 43
Table 3.10. Summary of Standard Deviations for the Author Computed Subsidence Values 44
Table 3.11. "Best" weighted displacements (84-88) 47 Table 3.12. Estimated Model Parameters (88-84) 48 Table 4.1. The Zenith Range Error due to Errors in Meteorological
Data (taken from Chrzanowski et al. 1988) 52 Table 4.2. Comparison of Subsidence Results - Levelling versus
GPS 62
Table 5.1. Preanalyses Results (standard deviations of subsidence in mm) 83
Table 1.1 Derived GPS and Field Levelling Data from Campaign No.1 105
Table 1.2 Derived GPS and Field Levelling Data from Campaign No. 2 106
Table 1.3 Derived GPS and Field Levelling Data from Campaign No. 3 (Tia Juana Section only) 107
vi
ACKNOWLEDGEMENTS
I would like to express my most sincere thanks to my supervisor, Dr. Adam
Chrzanowski, for his continuous assistance, encouragement and guidance and for his many
hours reviewing and discussing the manuscript and suggesting valuable ideas. Special
thanks are also due to Dr. Chen Yong-qi for his valuable scientific support and explanations.
My deepest gratitude to Mr. Juan Murria, technical advisor to the management of the
engineering and development department of Maraven. His motivation for the
accomplishment of this study and his endless interest in the scientific development of the
personnel of this department will always be remembered. A heartfelt thank-you to Jacinto
Abi Saab and Ladislao Jaeger, my supervisors within Maraven, for their unselfish support
and cooperation throughout the entire project, especially during the difficult times. Thank you to the scholarship program of Maraven S.A., which, under the leadership
of Carlos Rojas, has provided continual economic support and assistance. Debbie Smith is thanked for her kindness .and cooperation in making readable my symbolisms.
Finally, to my wife, Ingrid, my two children, Aura Christy and David Andres, and to
my parents Julio, Josefa and Selmira.J dedicate this effort for their love, understanding and
undying support.
vii
1. INTRODUCTION
Ground subsidence deformation has been a common hindrance to mining operations
and other engineering activities throughout the world. The most critical cases have been
usually connected with oil, gas, or water withdrawal. Typical examples are areas of
Wilmington in California, in the United States, Niigata in Japan, Mexico City in Mexico,
and the Costa Bolivar oil fields in Venezuela [Poland and Davis, 1969]. Many other cases
are discussed in Johnson et al. [1984]. In general, the problems.associated with ground subsidence may be summarized as
flooding, failure of engineering structures, devaluation of properties and reverse flow of
drainage systems. Consequently, there is an obvious need for the evaluation and prediction
of the deformation in order to minimize its impact on the surface environment.
In Venezuela, the first traces of subsidence deformation were detected in 1929 in the
area of Lagunillas, located on the Costa Bolivar Oil Fields, along the eastern coast of Lake
Maracaibo (Figure 1.1). The main cause of subsidence was reported to arise from the exploitation of relatively shallow (300 to 1000 m deep) oil reservoirs composed of highly porous and compressive unconsolidated material [Murria and Abi Saab, 1988]. The geomorphology of the area and its geographical location imposed serious limitations for
future development. A quotation from Kugler [1933] serves as a good example of this limitation: " ... subsidence was very bad, as you probably know, the wharf was
disappearing under the lake .. ". Under such circumstances a monitoring levelling scheme
was implemented in 1929. The collected information was, and still is, a source of valuable
information for managers and engineers involved in the development of the Costa Bolivar oil
fields. The information is generally used in the design of drainage master plans and coastal
protection dykes, oil exploitation and urban planning, calibration of subsidence prediction
LAG 0
DE
MARACAIBO
0 10
KMS
2
JUANA
LAGUNILLAS
MENE~> GRANDE .:: ...
. . .
. . .
. . .
. . .
. .
.. -.
. . . . oil fields
Figure 1.1. Relative Location of the Costa Bolivar Oil Field [after Puig, 1984]
3
models and most recently in the design of a contingency plan for the area in case of damage
to the protective dikes. The monitoring network has formed the basic vertical control for all
engineering projects in the area Up to the.present time, all. of the monitoring activities have been undertaken by
Maraven S.A., one of the major oil companies in Venezuela and a subsidiary ofPetr6leos de Venezuela S.A. (a state-owned holding company). Costly and slow geodetic levelling techniques have been used in the monitoring. Presently, the main surveys of the whole
inland subsidence area of about 1300 kffi2, consisting of 1624 benchmarks, are repeated at
2 year intervals with a portion of the network (fia Juana section which is about 1!3 of the total area covered by the network) being remeasured every six months for the purpose of checking the stability of off-shore platforms [Leal, 1987]. The main survey requires about 2 months for 6 survey crews to complete. A detailed description and evaluation of the
levelling scheme is given in Chapter 3.
Since 1984, as part of the efforts of Maraven S.A. to maximize productivity under the
implementation of modem technology, consultants from the University of New Brunswick
have been involved in the subsidence study to modernize and economize the present
monitoring scheme. Major improvements were sought by modifying the field techniques. Motorized trigonometric height traversing emerged as a possibility but it was soon turned
down as it provides little advantage over geodetic levelling in flat topographic conditions.
Differential satellite "Global Positioning System (GPS)" techniques, however, seemed to offer a feasible alternative. The commonly known advantages of GPS and claimed
achievable accuraCies led to a proposal for replacing the main levelling network by GPS
baselines in combination with lower order levelling surveys used for densification purposes.
The GPS network was meant to replace all connecting lines to stable areas and to adjacent subnetworks in other subsidence fields as well as to add information on the horizontal
behaviour of the deformation. As a first step, two test surveys were conducted on the Tia
Juana section of the main monitoring network in April 1987 and in October 1987 with
4
levelling surveys carried out at the same time to monitor subsidence of the offshore
platforms. Evaluation of the real performance and accuracy of GPS surveys under the
extreme climatic conditions of the Costa Bolivar oil fields was the main aim of the test
surveys. Results of these test surveys are discussed in Chapter 4.
Despite some difficulties, encouraging results were obtained from the two test
campaigns when comparing subsidence values obtained from GPS and from levelling
surveys. As a result, a full survey of the whole main network using GPS was conducted in
April 1988. Levelling surveys corresponding to the biannual monitoring campaign took
place at more or less the same time.
On the basis of this antecedent, the main objectiye of this thesis has been to design a methodology for integrating GPS and levelling surveys as a future survey scheme for
monitoring the subsidence in the whole area of the Costa Bolivar oil fields. The work has
developed around several tasks which are considered to be the contributions of the author.
They are listed as follows:
a) Choice of an adequate subsidence model; b) accuracy evaluation ofthe Maraven monitoring scheme; c) accuracy evaluation of the GPS derived height differences; d) development of a model for the integration of GPS with levelling; e) development of accuracy standards; f) development of a general field and computational strategy to implement the new
design;
g) economic analysis to study the feasibility of the new approach. Principles of the UNB Generalized Method of deformation analysis [Chrzanowski et
al., 1983; Chrzanowski et al., 1986] have been employed in the development of the
mathematical model for the integrated surveys and in the accuracy evaluation.
The discussion is outlined as follows. Chapter 2 conveys a general idea on the
aspects of subsidence modelling and describes in more detail the applicable model for the
5
case at hand. Chapter 3 is devoted to fully evaluating the present monitoring scheme.
Chapter 4 describes some aspects of the accuracy of the GPS derived height difference
and gives a brief evaluation of the GPS test results. Chapter 5 describes the problems
encountered in the combination of GPS with levelling and describes the mathematical model
and strategy to be used in the integration process. Finally, conclusions and
recommendations are given in chapter 6.
2. SUBSIDENCE DEFORMATION MODELLING
The evaluation and prediction of subsidence normally encompasses field monitoring
and modelling techniques. Within the general modelling techniques one can distinguish
between two distinct approaches - the physical modelling approach which considers the
physical laws and properties of materials involved in the deformation, and the geometrical
modelling approach based on the superficial geometry of the deformation [Vanfcek, 1987]. This chapter deals with the general aspects of geometrical modelling as applied to ground subsidence, within the context of the UNB Generalized Method of deformation
analysis [Chen, 1983; Chrzanowski et al., 1986]. A brief general background is first given, followed by a review of deformation models, the model-observations relationship
and final remarks.
2. 1 General Background
During the past few decades the geodetic community has directed new efforts into the
analysis of crustal movements and deformations in general. In 1954 the International
Association of Geodesy (lAG) appointed a special study group on crustal deformations and in 1960 established the Commission on Recent Crustal Movements [Pavoni, 1971]. In 1978, Commission 6 of the Federation Internationale des Geometres (FIG) created an ad hoc committee on the analysis of deformation measurements [Chrzanowski, 1981]. As a result, various modelling strategies and approaches into the analysis of deformations have
been developed. Very comprehensive reviews of modelling strategies for vertical crustal
movements (VCM) are presented in Holdahl [1978], Gubler [1984], and Vanfc'ek and Sjoberg [1987]. A theoretical review on different approaches to deformation analysis developed within the last decade, is given by Chrzanowski and Chen [1986].
7
Most strategies have been developed for modelling regional vertical crustal
movements based on scarce and heterogeneous data, such as: relevellings of national
geodetic levelling networks and small networks used in engineering and mapping projects, sea level variations, 'lake tilt data and detached relevelled segments.
The models have found wide application in local subsidence studies with the
advantage that the available subsidence data are characteristically more homogeneous and
abundant in the form of complete levelling networks (with no configuration defects), observed at regular time intervals and confined to short periods of observation.
Consequently, there is more flexibility for rigorous deformation analysis in the geometrical
interpretation of local subsidence deformation than in the regional VCM studies.
Although subsidence deformation is within the realm of VCM, a clear distinction
between the general objective of vertical crustal movements versus subsidence deformation studies must be made. Vertical crustal movement studies have been conducted to gain
deeper knowledge on the pattern and behaviour of fairly extensive areas and to interpolate or
extrapolate corrections to. homogenize observations gathered over,.considerable periods of
time. This is of particular interest to geodesists since it allows performance of simultaneous
adjustments of very extensive networks, for instance, the adjustment of national vertical networks or continental networks. Subsidence studies, on the other hand, are generally
conducted to evaluate the extent of man-induced subsidence in order to make decisions on
exploitation policies and planning, and on the design of engineering projects. In addition, most engineering activities in the affected areas are usually tied to the vertical geodetic
control defined by the subsidence monitoring networks which poses higher demands on the
analysis of the subsidence deformation, to guarantee the results with a greater degree of
confidence.
8
2. 2 Deformation Models
According to Chen [1983] and Chrzanowski et al. [1983] the deformation of a body
is fully described if the displacement field d (x, y, z; t- t0 ) is known. The displacement field can be approximated by fitting a selected deformation model to displacements
determined at discrete points
d(x, y, z; t-t0 ) = B(x, y, z; t - t0 )c (2.1) where dis the vector of displacement components of point (x, y, z) at timet with respect to a reference time fo,
B is a matrix of base functional values and
c is the vector of unknown coefficients.
The mathematical model (2.1) can be explicitly written as
( u(x, y, z; t-tcJ ) ( B J.x, y, z; t-t 0)Cu)
d = v(x, y, z;. t-tcJ = B v(x, y, : t-t0 )Cv w(x, y, z; t-t0 ) B .J.x, y, z; t-t0 )Cw
(2.2)
where u, v and w represent displacement components in the x, y and z directions
respectively. Since in su'bsidence studies we are mainly interested in tlie vertical component
(w or z) and since subsidence is generally independent of height, the general model for subsidence deformation can be reduced to
w(x, y; t-t0 ) = Bw(x, y; t-t0 )cw (2.3) which in short form can be written as:
w=Bc (2.4) where B is a row vector.
Different suitable functions may be used to approximate the deformation. A common
approach is to use algebraic polynomials.
Consider the general three dimensional polynomial n, fn nx j i k
w(x, y; t-t0 ) = L L x y (t-to) Cjik (2.5) k=l i=Oj=O
9
where nx, ny and nt are the maximum degree of the polynomial in the x, y and time
coordinates respectively, and cjik: is the polynomial coefficient with total number n = (ny+ 1 )(nx+ 1 )nt. Depending on the variations in nx, ny and nt, different models may be derived. Typical models using polynomials are given below.
a) Velocity surface model The model results from considering a linear deformation with time equivalent to nt=
1, nx and ny vary according to the spatial shape of the deformation.
For the deformation of one continuous block, equation (2.5) becomes ~ nx
w(x, y; t-t 0 ) = L L xjy \t-t 0 ) c j i . i=O j=O
(2.6)
Examples on the application of polynomial velocity surfaces may be found.in Vani~ek and Christodulidis [1974] and Vanf~ek and Nagy [1981].
b) Time-varying surface This model applies for the cases where the deformation is non-linear with time. The
model is of the same form as equation (2.5) but with nt > 1. An example on the application of this approach with additional considerations for episodic movements can be found in
Vanfcek et al. [1979].
c) Discontinuity model In the presence of discontinuities or to accommodate local anomalies, the area may be
divided into several'blocks and explicit models written for each block depending on its
behaviour. If for example, two Blocks A and B are considered (Figure 2.1a), where all points in B moved together and linearly with respect to A during the interval (t-t0 ) (relative rigid body movements), the model will be
wA(t)=O wB(t) = (t-tJHB (2.7)
10
where subscripts A and B represent the points on the block A and B, respectively, and H
will be the velocity of vertical movement and equivalent to c in the above equations.
If for instance, the blocks experience linear temporal deformation within themselves
(Figure 2.1b) as well as relative body movement, the model for all points in each block will be of the form:
(2.8)
where point x0 y0 is a reference point. Different combinations of all the above cases may be
used depending on the following factors: the desired accuracy of the modelling, the
redundancy of the observations, the number of available epochs, and the distribution of the
data.
An alternative approach in some cases has been the use of multiquadric analysis
(Hardy [1978], and Holdahl and Hardy [1979]), whereby the polynomial is replaced by a suitable quadric form. According to Holdahl and Hardy [1979] this has the advantage of
producing more appropriate automated graphic representations of the subsidence at extreme
values outside the data area.
For the case of the Costa Bolivar in Venezuela, the subsidence monitoring network
constitutes the basic vertical control for all the engineering activities undertaken in the area.
Therefore, a knowledge of the subsidence of each individual benchmark is the immediate
goal of the subsidence study. Spatial modelling obtained through a surface fitting, although
suitable for most general purposes, will not provide the most accurate elevations at discrete
points, especially if one considers the irregular shape of the Costa Bolivar subsidence
deformation (see Figure 2.2 below). Thus, the subsidence values for each individual
11
z
a)
z
A B
~------~------~---r
b)
X
Figure 2.1. Examples of deformation with discontinuity
12
479
1329 Op
Contour interval 0.5 m
6914
Figure 2.2. Subsidence Basins (Cumulative Subsidence 1926-1986)
13
benchmark must be modelled and derived first. Later on, any desired surface fitting to the
vertical displacements may be performed either analytically or graphically in order to obtain a
graphical representation of the subsidence basin.
As it will be discussed in Chapter 3, the subsidence along the Costa Bolivar, at least
within a time span of a few years (approximately 10 years), seems to follow a linear trend. Therefore, each particular benchmark, at least initially, may be considered as a rigid block
undergoing linear displacement in time with respect to a stable block represented by the
benchmarks located in a stable area. The constant velocity model, equation (2.7), has been selected for the subsidence modelling which is discussed in more detail in the next section as
well as in Chapter 5.
2. 3 Discrete Point Constant Velocity Model
In the case of subsidence monitoring studies, most observables fall under two general
types: either height difference or tilt observations. They encompass all the geometrical data
such as: tide gauge observations, relevellings, direct tilt measurements and spatial position
changes.
From the principles of the generalized approach to deformation analysis
[Chrzanowski et. al., 1983], it follows that for the model estimation, the relationship
between the deformation model and the observables could be established through the general
equation
(2.9a)
(2.9b)
where t(ti) is the vector of observations in epoch ti (i = 1, 2, ... , k) ~ is a vector of unknown quantities, which may be the coordinates or expected values
14
of the observables or the combination of both at reference time t0
A is a transfromation matrix from s to t and v is the vector of residuals
B'i is constructed'from the matrix B of deformation model (2.1) relating the unknown coefficients to the change in s.
Thus, for a levelled height difference between any two points Pk and Pj at epochs (t0 ) and (t1) the general observation equations may be written in the form
(2.10a)
(2.10b) Considering the constant velocity model equation (2.7), the general observation
equations (2.10) may be rewritten as
(2.11a)
(2.1lb)
. .
where the point velocities Hj and Hk are elements of the vector c of the model parameters to be estimated, and Hj(t0 ), Hk(to) are elements of the vector of unknown constants s.
In the case of the subsidence studies of the Costa Bolivar, the only observables used
in the subsidence surveys are height differences of individual levelling or GPS lines.
Therefore, only the model expressed by equations (2.11) will be discussed. I"J
Taking AiB'i = Bi in the general equation (2.9), a (k+ 1) multi-epoch solution may be expressed in matrix form as:
15
+ = (2.12)
v(t~
If only point velocities are desired, ~ could be treated as a vector of nuisance parameters which can be eliminated in the process of the least squares estimation of the c
parameters using the well known elimination methods. On the other hand, if one is
interested in estimating a set of homogeneous heights at a chosen reference epoch (t0 ) together with the solution of c, then, both ~ and c will form part of the estimated parameters
in the adjustment. Another approach to estimate the unknown coefficients c is to rise the differences de
in a two epoch-comparison. "In this case, equation (2.9b) is subtracted from (2.9a) if the observables in two epochs are identical, i.e., Ai = A0 , and the following equation is
obtained:
d~(L1ti-o) + dv(L1ti-o) = AiB'ic (2.13) where .:1ti-o is the time interval between any epoch ti and the reference epoch t0 . The height
differences observation equations (2.11) can then be re-written as: MHkj(L1ti_o) + dvkiL1ti_o) = (ti-toXHrHJ . (2.14)
The general solution of all the previously discussed cases can be achieved through the
application of the least squares criteria. For details on the estimation process and selection
of model parameters, the reader is referred to Chrzanowski et al. [1983] and Chrzanowski et
al. [1986].
16
One can adjust the observations for each campaign separately in a static mode and then fit the deformation model to the derived displacements. This has the advantage of
allowing data screening and statistical evaluation, as well as trend analysis for the
appropriate selection of the deformation model. However, a major limitation is that significant deformation may take place during the data collection period within each survey
campaign. Therefore, the kinematic adjustment case discussed above is found to be the most appropriate, especially where large subsidence rates are expected.
In general, the discussion has relied on various assumptions which have been
implicitly made and are listed as follows:
i) For the case of discrete point models, it has been assumed that the observations correspond to a complete network with no configuration defects. Otherwise the
existence of detached segments will cause singularities in the solution.
ii) At least two observation campaigns of the same network geometry have been assumed to exist.
iii) The gravity variation in the area has been assumed to be sufficiently small to allow observed (levelled) height differences to closely approximate the corresponding geopotential or orthometric difference [Hein, 1986].
2. 4 Remarks on Other Approaches
For the sake of completeness another less common approach to modelling, referred to
as the stochastic approach [Hein, 1986], has also been used in vertical crustal movements. This approach is based on the method of least squares collocation. The deformation is
segregated into three basic parts: a global trend, a regional signal and the noise which
includes measuring errors and the individual movements of the benchmarks On this basis the
observation equations are set up and solved using the approach of Moritz [1972] as referred
to by Hein and Keistermann [1981]. Hein [1986] compares a so-called "mixed" model using this approach against a combined point velocity- multiquadric model showing slight
17
advantages in the results obtained with the mixed model and a major drawback in the error information given by the multiquadric approach. A more general approach to include a
wider variety of geodetic data into modelling by this technique is discussed in Hein and
Keistermann [1981]. Another approach may be the use of splines but very little has been done in this respect. Additional discussion on the subsidence modelling is presented in
Chapter 5.
3. EVALUATION OF THE PRESENT MONITORING SCHEME
This chapter is intended to convey a clear picture of the state of the Maraven
subsidence monitoring scheme presently in use, beginning with a brief historical synopsis
and general description of the existing monitoring network and computational technique
used. It touches briefly on field procedures, discusses economic aspects and presents a
fairly complete accuracy evaluation of the last three campaigns.
3.1 Historic Synoosis
The oil extraction in the Costa Bolivar oil fields (see Figure 1.1) began on a small scale in the field of Mene Grande in 1914, followed by Cabimas in 1922 and by commercial
exploitation in the field of Lagunillas in 1926 [PDVSA, 1984]. According to Collins [1935], the land adjacent to the village ofLagunillas was mostly swamps and marshes that required the development of a drainage system prior to the development of the oil fields.
Trutmann [1949] reports that in 1927 a levelling survey (swamp survey) was conducted for preliminary drainage studies in the area by the Topographical Department of the Venezuelan
Oil Concessions Company Ltd. (V.O.C.), part of the Shell Caribbean Consortium in Venezuela. Later on in 1929 the observation of permanent flooding in the production areas
raised the suspicion of subsidence in the field, which according to Trutmann [1949] was confirmed by a check on the swamp level survey of 1927 showing subsidence values of the
order of 42 em. This was cause of general alarm and lead to the immediate implementation
of a preliminary monitoring scheme. Long connecting lines to the supposedly stable areas
were established, and after several campaigns by the middle of 1934 a subsidence rate of
20 em/year was confmned [Trutmann, 1949].
18
19
As exploitation continued to expand into neighbouring areas over land and offshore,
expansion of the monitoring surveys became necessary. Monitoring began in the area of
Tia Juana and Bachaquero in the years 1937 and 1938 respectively (taken from the subsidence records available at Maraven S.A.). During a few years, from 1934 to 1942, monitoring was generally carried out annually. After 1942 the surveys were spaced at
intervals of two years. It is believed that in the early 1940's the whole monitoring scheme
was redesigned, since the VOC company took over from Creole the responsibility for the
offshore subsidence monitoring. The offshore subsidence has been monitored by means of
water level transfers to well platforms using temporarily installed tide gauges [Leal, 1987]. As time went on, three subsidence basins (polders) have developed above the areas of
major exploitation, as depicted in Figure 2.2, where the contour lines represent cumulative subsidence. Consequently, and in response to the requirements of reservoir and
construction engineers, the monitoring network has been further expanded and densified.
Presently, there exists a main monitoring network covering the fields of Tia Juana,
Lagunillas and Bachaquero, and two smaller subnetworks connected to the main network
and located in the fields of Cabimas and Mene Grande, whose geographical locations are
shown in Figure 1.1.
3. 2 Network Description
The main levelling frame is shown in Figure 3.1. It covers a geographical area of
about 1300 km2 and consists of 618.9 km of first order class II (U.S. specifications) levelling lines of which 167.3 km are used for connections to the assumed stable area.
Within the network itself there exists an array of second order class II (U.S. specifications) levelling lines for densification. Figure 3.2 shows in detail a small section of the network illustrating the pattern followed by these second order lines. This pattern is
denser in the areas of larger subsidence rates which correspond to larger exploitation zones
near the centers of the main "polders" shown above. The total length of the second order
479
TIA JUAIIA
L A6UIIILLAS
LAKE MARACAIBO
8ACHAOUHO
20
Figure 3.1. Main Levelling Net.
0
Main Lines
Nodal Lines
21
277A
218 217 A 239A
13664 'l--+-----"'r103
1367
1018
1--20A
l AK MARACAIBO
--- r.tAIN lVfLLIN6L INES
SECO"OUy "'- "OOAL liH5
ACCSS AOA 0~
II'INER 0 IK$
Figure 3.2. Detail showing Secondary and Nodal Lines
22
levelling lines adds up to 553.7 km. Additionally, the two subnetworks in Cabimas and
Mene Grande also consist of ftrst and second order lines which add 160 km of frrst order
lines and 67.3 km of second order lines. The levelling lines which connect both
subnetworks to the main network total68.9 km.
The whole monitoring network, including Cabimas and Mene Grande, consists of
1624 bench marks (BM's), from which two types of monuments could be distinguished: the deep BM's located mainly along the connections to the stable areas (20 - 30 km inland) and anchored to a depth of 30m, and shallow BM's used for densification purposes and
connections to the subnetworks. The shallow BM's are cast in concrete inside steel pipes to
a depth of approximately 1.7 m. The average spacing between BM's in the network is
approximately 400 m.
The offshore subsidence is monitored through an array of 306 well platforms. For
the purpose of the analysis herein, only the inland network is considered. A summary of the
network characteristics is given in Table 3.1.
3. 3 Field Procedures
A total of about one month is needed by 6 survey crews to survey the first order
framenet. In order to minimize the accumulation of temporal heterogeneities that could
contaminate the observations due to the dynamic behaviour of the subsiding surface, all
survey crews work simultaneously starting from outside the subsidence basins toward the
areas of maximum subsidence. The field levelling procedures follow closely the
requirements outlined in the NOAA [1984] standards and specifications for the U.S. ftrst
order class II geodetic control networks. The only exceptions are that temperature gradients
are not measured for refraction corrections since the area is mostly flat and the effect of
refraction is expected to be greatly minimized by balanced lengths of sight . Gravity
measurements have not been taken either. The instrumentation used includes Wild N-3 and
NA2 and Zeiss Ni2 levelling instruments, with parallel plate micrometers and invar
23
Table 3.1. Summary of Costa Bolivar Network Statistics.
DESCRIPTION
Total km of first order levelling lines
Total km of second order levelling lines
Total km of levelling lines connecting to stable areas
Total km of levelling lines in connections to main network
Total number of BM's in connecting lines
Total number of BM's
Area covered (km2)
MAIN NETWORK
618.9
553.7
167.3
205
1436
1296
CABIMAS SUB-NETWORK
97.6
38.5
18.4
8
102
66.5
MENEGRANDE SUB-NETWORK
62.5
28.8
50.5
37
86
24.8
rods with one or one half centimetre divisions. The second order levelling is performed
according to the U.S. standards for second order class II surveys. The same
instrumentation as in the first order levelling is used. Measurement of the second order
lines takes approximately one month when using 4 survey crews. No specific measuring
pattern is followed with the exception of the "nodal lines" which are the lines connecting the
main network to specific junction BM's. The nodal lines are measured simultaneously by several crews since they are generally located at places where larger subsidence rates occur.
The same procedures are used in the survey of the two subnetworks, Cabimas and Mene
24
Grande, which require about one month time with one levelling crew. HP 41CV calculators
have been used as field data collectors during the last three campaigns, increasing the speed
of the field work.
3. 4 Data Processing Technique
The basic principles of the data processing method which is described below are
believed to have been in effect since the early monitoring times. The general computational
sequence presently used at Maraven S.A. is outlined in flowchart form in Figure 3.3. Each
step will be briefly explained excluding the computation of the subnetworks of Cabimas and
Mene Grande, in lake (offshore) subsidence and graphical representation. The method will be referred throughout this thesis as the Maraven method.
3.4.1 Computation of datum lines
The monitoring network is connected to the assumed stable area through three
connecting lines consisting mainly of deep BM's. They are called "datum lines" since they
provide the fixed constraints for the network adjustment (see Figure 3.1). The elevations are computed by the following procedure [Shell, 1954]:
(a) A set of provisional elevations is computed for the deep BM's on each "datum line" starting from the elevation obtained in the previous campaign for each extreme BM
farthest inland and adding algebraically the averaged height differences observed.
(b) The sum of the provisional elevations of all the deep BM's in each line is compared with the corresponding sum of the elevations in the previous year and the differences
are computed.
(c) Based on the assumption that the deep BM's remain gempletely sta~Jlf!l, the tlifferens@s from (b) are divided by the number of deep BM's in each line. The definitive elevations are finally obtained by adding the estimated correction to the provisional
elevations in (a).
! COMPUTATION
OF S U B N E T W 0 R KS
I
25
PRE- PROCESSED
FIELD DATA
COMPUTATION OF
DATUM LINES
ADJUSTMENT OF MAIN
LEVELLING NETWORK
COMPUTATION
OF NODAL LINES
COMPUTATION OF
SECONDARY LINES
COMPUTATION OF FINAL HEIGHTS AND SUBSIDENCE
GRAPHICAL REPRESENTATION OF SUBSIDENCE
FINAL REPORTS
COMPUTATION OF
LAKE SUBSIDENCE
Figure 3.3. Computational Sequence Flow Chart
26
The new elevations are then considered as fixed for the network adjustment. Thus, in each campaign a new datum is created. Table 3.2 shows the elevations of the extreme BM's in
each line for several campaigns. Notice the shifts introduced by this procedure especially on
BM's 1175 DP and 1329 DP which are supposed to be the most stable points in the network
since they are located farthest inland. This shows that practically no BM is actually
considered stable between campaigns and the absolute elevations of all points in the network
will be systematically affected. This is further discussed in section 3.6.3.
Table 3.2. Elevations of Reference Bench Marks [m].
BM 1980 1982 1984 1986 1988
1329DP 99.512 99.512 99.511 99.510 99.519 1002DP 59.508 59.510 59.510 59.511 59.507 1175 DP 53.316 53.320 53.316 53.309 53.302 185DP 31.134 31.131 31.135 31.155 31.176 1326DP 54.628 54.629 54.628 54.628 54.628 1324DP 40.364 40.364 40.365 40.366 40.364
3.4.2 Adjustment of main levellin& network The solution for the first order lines is attained through the least squares adjustment of
the main levelling network using condition equations. Twelve independent condition
equations are formed as shown by the Roman numerals in Figure 3.1. Constraints are
enforced through condition equations IX and XII where the elevations for BM's 1002 DP,
185 DP and 1326 DP, which were computed using the aforementioned procedure and are
located on each datum line, are to be treated as fixed. A weight corresponding to the
inverse of the length [km] is given to each line. The normal equations are solved using the
method of correlates and the solution estimated through the application of the Gauss-Doolitle
method [Rainsford, 1957]. The whole network is adjusted in a static mode. The estimated elevations have been time tagged (for the last 20 years) to the first of March of the year of
27
the survey campaign, since it is generally the average date of the main network survey. No
error analysis is petformed apart from the computation of loop misclosures and a posteriori
variance factor to indicate the global quality of the observations.
3.4.3 Computation of nodal lines
A group of 3 or 4 nodal lines connecting the main network to a particular nodal BM
is called a node (Figure 3.2). Each node is computed by simple extrapolation in time of the adjusted main network BM's at each connection to the date of the survey of each line. A weighted elevation for the junction BM is computed, and then each line is adjusted accordingly.
Interpolation of all of the elevations to the reference date of the main network takes
place using the subsidence rate obtained from the previous campaign, for each BM along the
line. Residuals (weighted minus observed elevation at the junction BM) larger than 2.8 mm ..Jk, where k is the distance in kilometres, lead to the rejection of the particular nodal line. Rejected lines ,are usually remeasured in the field. There are nine node cases in the main network as shown in Figure 3 .1.
3.4.4 Computation of secondary lines
Secondary lines are those of the second order accuracy which are connected either to
the main reference network or to the nodal lines. The computation follows a very similar
procedure as that for the nodal lines. Time extrapolation of the elevations of the two end
BM's to the date of the survey of the line is used to compute a height difference discrepancy
for each line. Then, using the same rejection criteria as above, the line is either accepted or rejected for remeasurement. Once accepted, the line is adjusted and interpolated in time back to the reference date.
28
3.4.5 Additional remarks
In 1984, as part of an automation project, the whole procedure described above was programmed into a PDP 1170 minicomputer. The automation project also included field data collection .with the HP41CV calculators, transfer of data and pre-processing using an
HP85 microcomputer, and transfer from the HP85 to the PDP1170 for final processing.
The computational procedure is still slow and tedious due to the inflexibility of the existing
software and obsolescence of the methodology. A total of 3 to 4 weeks is normally needed
to process the whole data.
3. 5 Economic Aspects
Monitoring has always been performed by precisiongeodetic levelling techniques as
described earlier, which is a slow and costly operation.
The major costs involved in the present inland monitoring scheme arise from the three main sources: field levelling work, bench mark maintenance, and supervisory plus data
processing activities. This. section is intended to develop approximate relationships to
estimate the costs of each one of these activities based on previous experience gained by the
author as a manager of the last two campaigns (1986 and 1988). The values are by no means exact since approximate cost rates have been used and minor costs have been
neglected for simplicity. The costs of post-processing for the elaboration of final contour
maps and monumentation reconstruction or replacement are not included.
3.5.1 Cost of levelline
Performance in geodetic levelling is directly related to the field procedure and existing
meteorological conditions. High temperature and humidity generally limit the sight lengths
and observation hours. Although a prevalent average temperature of 3rC and 80%
humidity is encountered in the Costa Bolivar oil fields, an average daily performance of 6
km has been experienced with survey crews consisting of one surveyor and four
29
non-qualified labour workers. Second order procedures can generally be considered the
same as first order but single run. Therefore, the general relationship to estimate the cost of
levelling per kilometre for day shifts of 8 hours may be as follows:
Cost lev/km = 1.33 hr/km (4CA + CB + C1) (3.1) where CA and CB are the costs of non-qualified and technical labour per hour respectively,
and C1 is the cost of instrumentation per hour, which includes vehicle and surveying
instrumentation. Assigning values to the above variables in Bolivares (Bs) which is the currency in Venezuela, of CA = 150 Bs/hr, CB = 200 Bs/hr and C1 = 100 Bs/hr the cost of
one kilometre of levelling would be in the order of 1200 Bs/km, which is equivalent to about
100 Canadian dollars per kilometre using the present exchange rate of 14.50 Bs/US$
applicable to oil industry operations and a ratio of 1.20 Cdn.$/US$.
3.5.2 Cost of maintenance
The maintenance of BM's mentioned here consists basically of minor repairs (e.g. painting) and vegetation trimming for each BM prior to the surveys. An average performance of 12 BM's per day per crew of 4 workers has been maintained over the past
years. The following relationship can be used to obtain the maintenance cost/BM:
Maintenance cost/BM = 0.67 BM/hr (4CA + C1) (3.2) where C1 now includes the cost of vehicle and working tools per hour.
Using the same approximate values as before with C1 again equal to 100 Bs/hr (since it includes cost of materials), the cost of maintenance per BM is computed to be 469 Bs/BM, equivalent to 39 Cdn.$ per BM.
3.5.3 Cost of supervision and data processing
Costs related to data processing and supervision normally involve the performance of
two surveying engineers. One dedicated entirely to supervisory duties, planning, logistics
and administration, and the other concerned with daily data logging and processing.
30
Although the cost of the former would generally be higher, an average daily rate Cp = 2000
Bs/day for each could be used. This is equivalent to 166 Cdn.$ per day.
3.5.4 Total estimated cost of one campaign
On the basis of the above figures and considering a total of 1624 BM's, 1469 km of
levelling lines and 240 days for supervision and data processing, the total cost of one
campaign may be established. Note that this total cost includes neither the costs of post
processing for the elaboration of final contour maps and monumentation nor the cost of
offshore subsidence surveys.
The total cost may be estimated as follows:
Cost of levelling
1469 km x 1200 Bs/km ..... .
Cost of Maintenance
1624 BM's x 469 Bs/BM ....
Cost of supervision and data processing
240 days x 2000 Bs/day ....
TOTAL COST
1,762,800 Bs
761,656 Bs
480.000 Bs
3,004,456 Bs
This is equivalent to 248,644.6 Cdn.$ using the same exchange factors as above.
Notice that the major cost arises from levelling.
3. 6 Accuracy Evaluation
As already mentioned, the described computational technique has not provided
sufficient information and flexibility for a proper assessment of the results. Therefore, an
independent evaluation of the actual accuracy of the subsidence monitoring scheme has had
to be performed by the author. The data of the last three survey campaigns, which took
place in 1984, 1986, and 1988 have been used in the accuracy analysis employing the
MINQE technique mentioned below.
31
3.6.1 Description of survey data
As mentioned earlier, survey data of three previous campaigns was available for the
accuracy analysis. The same network geometry was kept during campaigns with the
exception of a few BM replacements. Athough most of the 1986 and 1988 data was
available, only the first order levelling data from the main levelling network and a section of
the second order densification data (shown in Figure 3.2) in the Lagunillas basin was selected for testing. For the 1984 campaign, only the first order levelling data of the main
network was available. Total height differences of the levelling lines between main junction BM's were taken for the analysis. Table 3.3 shows a summary of the data.
Table 3.3. Statistical Summary of Survey Data.
Description 1984 1986 1988 of Data
First order lines 36 49 49
Second order lines 26 26
Total lines 36 75 75
Number of bench marks 26 50 50
3.6.2 Accuracy of levelling surveys
Geodetic levelling is affected by two types of errors - random and systematic.
Random errors are always present in the measurements and cannot be eliminated.
Systematic errors, however, could be eliminated or minimized by proper field procedures
and calibration. A concise review of the characteristics and methods to eliminate most of
32
these errors can be found in the manual for geodetic levelling from the U.S. National
Geodetic Survey [NOAA, 1981]. Since 1912 with the introduction of the Lallemand formula, many models to combine
the effect of random and systematic errors in levelling have been suggested; the main
disagreement being on the interpretation of the systematic effect [Wassef, 1974]. On the other hand, correlations within and between neighbouring lines also exist and has been
researched by several authors [e.g., Vanfcek and Grafarend, 1980]. Some of the suggested models contain parameters which are empirical and too subjective making their evaluation and application rather unrealistic in the situation at the Costa Bolivar. A more rigorous and
practical method, recommended by Chen [1983], is to use variance components estimation techniques to evaluate model 'parameters from field data. The Miriiimim Norm Quadratic Estimation (MINQE) described in Chen [1983] has been successfuly used by Chen and Chrzanowski [1985] for estimating error model parameters in levelling networks. The
MINQE technique is part of the UNB Generalized Method and was used here by the author in the evaluation of the Venezuelan levelling data. The simple model cr2e = cr2ik (where k is distance in km and O'i is the standard deviation per kilometre) was used in MINQE to evaluate the components cri corresponding to the first and second order data.
The systematic effects were considered minimal since the levelling lines are rather
short(- 10 km) and the area is mostly flat. The estimated variance components in the form of standard deviations together with their corresponding standard errors for all the
campaigns are shown in Table 3.4. Of course, in single variance estimation there is no need
for sophisticated variance estimation techniques since the a posteriori variance factor
estimation will be sufficient. However, to separate the variances corresponding to
heterogeneous data, it becomes necessary. In the combination of first and second order data
for the 1986 and 1988 campaigns, the same model was used but with cr21 and cr211
representing the variances corresponding to the first and second order data respectively.
The estimated standard deviations are also shown in Table 3.4.
33
It is necessary to point out that neither gravity nor any other corrections were applied
to the data. Thus, the misclosures are contaminated by the net effect of neglecting gravity
corrections, rod calibration errors, residual refraction, and other error sources affecting the
measurements. It has been common practice at the Costa Bolivar not to apply any
corrections to the levelling data. Thus, the estimated accuracy reflects the real levelling
accuracy used by the Maraven computational method.
Table 3.4. Levelling standard deviations as determined by MINQE
EPOCH
1988
1986
1984
Single parameter estimation
( rn)
34
Cumulative Subsidence BM A
0 ~--------------------------------
- 1
..___ __
-2
'~ ~.._.__ ...................
-4 -...
oO 000o 0 o 0 " Oo 0 00o0o 0 o00 000000 OOhOh- 00 OOoOOoo=~:~ oOOOOOOOoO OOOOoOoO
-5 Oooo ooooOoooO ooOoOooooOOOoOO hOO OOoo 0 0 00 -
- t) f-----L-_L__L__L_L_jl__L_L_j'---+--'-' ---''-'--'-------1..1. 1qno
.._L 1Dt10 1960
Year Figure 3.4. Historic Plot of Critical Bench Mark A
(Lagunillas basin)
~""'' 'lil, ...
"1980
( rn)
35
Cumulative Subsidence BM 608
0 -----
--2 -
-3 ------
-4 .............................................................. .................. ~ .. .,.,'-,.
--...., .........
..........
I ..J.._.....l_.1..__.1__J__L_L......JIL...LI -+1-'L...LI --'-----'l___l__l__,__l -'-'-''-++-'-'--'-' -'---i.-l__.__. _ _ji>-J.._I '. I ___ L -5r-- - r-
1940 1960 1980
Year Figure 3.5. Historic Plot of Critical Bench Mark 608
(Bachaquero basin)
1
""8
( rr1)
36
Cumulative Subsidence BM 215
--------------------------------C) ----------
~--~
-~--
~ -2 --
,, - '--1 --
-4
-5
- 6 1---'-~~..L.......L..~I__.I I I l I I I I I I I I 1920 1940 1960
Y'ear Figure 3.6. Historic Plot of Critical Bench Mark 215
(Lagunillas basin)
I I I 1980
37
3.6.3 Validity of the static network assumption
The dynamic characteristics of the subsidence introduces systematic heterogeneities
into the data since the survey is not performed at one instant of time. Although the field
surveys are planned in such a way as to minimize this systematic noise, it is necessary to
investigate the significance of the static network assumption for the main network
adjustment. Although the subsidence in Lagunillas and Bachaquero does not seem to exhibit linear behaviour in time as depicted by historic plots of several BM's located over the
major subsidence basins (Figure 3.4, 3.5, 3.6), a short term linear behaviour, up to a decade or so, can be safely considered as valid for the choice of a simple velocity function in
a kinematic adjustment of the network. In this investigation the three levelling campaigns were compared using first a separate campaign adjustment in the static mode and then the kinematic modelling approach discussed in Chapter 2.
Parametric adustments with minimal constraints holding Point 1175 DP fixed were
performed. The previously estimated variances (from the MINQE method) were used for weighting the observations. The subsidence computed through both methods for the most
critical BM's is shown in Table 3.5. No significant difference between both methods is
observed, leading to the conclusion that, at the present rate of the subsidence and provided
that the surveys in the past were performed simultaneously towards the areas of the
maximum subsidence, the assumption of the static adjustment has not introduced significant biases in the subsidence and elevations determination.
38
3.6.4 Stability of the reference network
A very important aspect of deformation monitoring is the proper assessment of the
stability of the reference network. Distorted displacements may lead to erroneous analysis
and interpretation of deformations. Over the last few years this topic has been fully
investigated Several methods have been developed within the activity of the FIG committee
on the analysis of deformation surveys [Chrzanowski and Chen, 1986]. One of the
Table 3.5. Comparison of Subsidence Determinations.
Subsidence from Static Adj. Subsidence from Kinematic Adj. (mm) (mm)
Bench 84/86 86/88 84/88 84/86 86/88 84/88 Mark
185DP +26.4 +28.5 +54.9 +26.3 +28.5 +54.8 1056 -38.4 -45.6 -83.9 -37.9 -45.5 -83.8 329B -42.6 -34.0 -76.4 -42.5 -34.0 -76.6 -AB -116.6 -83.9 -200.3 -115.9 -83.9 -199.8 856 -25.5 -31.5 -56.2 -24.3 -31.9 -56.0 846A -61.6 -44.5 -105.1 -60.1 -45.0 -105.0 639A -14.8 -14.2 -28.0 -14.0 -14.7 -28.3
methods which is part of the aforementioned UNB Generalized Method is the iterative
weighted similarity transformation. The method is meant to yield the "best" relative
displacements following an iterative procedure to minimize the first norm of the estimted
displacement vector as described in Chen [1983], and Secord [1985]. The method has been applied by the author to analyze the stability of the reference BM's in the monitoring
levelling network.
The reference BM's are located along the aforementioned "datum lines" which have
been used to constrain the adjustment. The "best" displacements and their significance as obtained from the application of the weighted similarity transformation to different epoch
39
combinations is shown in Table 3.6. Only the extreme BM's on each datum line are listed.
A non-iterative procedure especially for levelling networks presented in Chen et al. [ 1988] has been followed by the author using his own software.
Table 3.6. "Best" Weighted Displacements for Reference Bench Marks
BM 86-88 Significance 84-88 Significance 84-86 Significance [mm] level [mm] level [mm] level
1329 DP 8.9 0.79 6.4 0.40 3.0 0.22 1002DP -3.5 0.85 4.1 0.34 13.1 0.86 1324DP 9.6 0.61 9.4 0.60 4.7 0.29 1326DP 12.4 0.70 11.2 0.64 3.7 0.22 1175 DP 7.8 0.58 14.8 0.88 12.6 0.73 185DP 36.3 > 0.99 69.7 > 0.99 39.0 > 0.99
The results show a significant uplift of BM 185 DP which is responsible for the
apparent subsidence of reference BM 1175 DP of the Lagunillas datum line when using the
earlier described datum lines computation method.
A shift of -7 mm to BM 1175 DP was introduced in the original calculations at
Maraven for both the 84-86 and 86-88 comparisons, as revealed by the different elevations
estimated for this BM in 1984, 1986 and 1988 (see Table 3.2). However, the author's results which are shown in Table 3.6 do not indicate any significant movement of that BM.
The same applies to BM 1329DP which was shown in Table 3.2 as having a movement of
+9 mm between campaigns 86-88. The author's calculations show again that its movement
is statistically insignificant (see Table 3.6). It can be concluded that, although the most distant BM's inland which correspond to
the ends of the three datum lines seem to be stable, the Maraven computational method
introduces systematic shifts to some BM's which are actually stable. This can lead to false
elevations and misleading subsidence results. Fortunately, since the Maraven method uses
40
an overconstrained adjustment (section 3.4.2) the smoothing effect that takes place decreases the total effect of the falsely introduced movements. The worst results are
expected when the same or similar shifts are introduced at least at two of the reference
BM's.
3.6.5 Final accuracy of elevations in single campaigns
Since the Maraven computational scheme does not provide stochastic information to
assess the accuracy of the results, an equivalent static parametric adjustment for each campaign was performed. The previously estimated variance components were used and the
extreme BM's on each "datum line" (1175 DP, 1329 DP and 1326 DP) were held fixed. Table 3.7 shows a comparison between the elevation values for the same campaigns
obtained at Maraven and the new values obtained by the author. Obviously, the differences
show the systematic effect of the datum shifts in the Maraven calculations as discussed in the
previous section. The systematic trend is equivalent to -7 mm for the 1986 and 1988
campaigns and to about -3 mm for the 1984 campaign.
A maximum standard error of 7.7 mm which is equivalent to 15 mm at a 95%
confidence level was obtained for the adjusted elevations (see Table 3.8). This is, of course, datum dependent as the elevation errors increase with the distance from the
constrained points.
It can be concluded that the total uncertainty in the absolute elevations as obtained
from the Maraven computational method may reach 15 to 20 mm at the 95% confidence
level.
3.6.6 Accuracy of the subsidence determination
The accuracy of the subsidence determination was derived from the separate
campaign adjustments. The results of 1984 and 1988 give a maximum standard error of 10.9 mm which is equivalent to 22 mm for absolute subsidence at the 95% confidence level
41
Table 3.7. Comparison of Elevations.
Elevations by Maraven
Elevations by Author
Discrepencies (Maraven-Author)
Bench Elev. Elev. Elev. Elev. Elev. Elev. A[mm]A[mm]A[mm] Mark 84 [m] 86 [m] 88 [m]) 84 [m] 86 [m] 88 [m] 84 86 88
1002DP 59.510 59.511 59.507 59.506 59.513 59.501 +4 -2 +6 185DP 31.135 31.155 31.176 31.134 31.165 31.192 +1 -10 -16 744 48.941 48.938 48.936 48.937 48.941 48.932 +4 -3 +4 734 32.704 32.736 32.702 32.703 32.740 32.702 +1 -4 0 1056 31.221 31.183 31.140 31.221 31.188 31.141 0 -5 -1 411 25.056 25.069 25.063 25.054 25.074 25.063 +2 -5 0 387 4.282 4.270 4.270 4.282 4.275 4.271 0 -5 -1 329B 0.817 0.775 0.741 0.817 0.780 0.745 0 -5 -4 1390A 1.314 1.300, 1.291 1.315 1.307 1.297 -1 -7 -6 1703 14.897 14.981 14.877 14.899 14.898 14.885 -2 -7 -8 M 17.420 17.414 17.405 17.424 17.423 17.417 -4 -9 -12 184A 27.517 27.519 27.515 27.516 27.529 27.529 +1 -10 -14 1791 10.521 10.511 10.501 10.525 10.519 10.513 -4 -8 -12 -AB 1.095 0.980 0.892 1.098 0.987 0.901 -3 -7 -9 117 2.178 2.168 2.164 2.181 2.174 2.172 -3 -6 -8 46A 5.288 5.274 5.270 5.291 5.280 5.278 -3 -6 -8 856 14.651 14.629 14.594 14.654 14.635 14.601 -3 -6 -7 1725 61.511 61.503 61.505 61.512 61.508 61.511 -1 -5 -6 846A 10.006 9.948 9.900 10.008 9.954 9.906 -2 -6 -6 546A 3.591 3.586 3.573 3.594 3.592 3.581 -3 -6 -8 639A 8.229 8.220 8.202 8.231 8.224 8.207 -2 -4 -5 691B 0.752 0.730 0.721 0.755 0.734 0.726 -3 -4 -5 1324DP 40.365 40.366 40.364 40.367 40.367 40.365 -2 -1 -1
42
Table 3.8. Summary of Standard Deviations (o)for Author Elevations.
Bench cr[mm] cr[mm] cr[mm] Mark 84 86 88
1002DP 4.5 3.2 4.5 185DP 4.2 2.9 4.2
744 4.7 3.3 4.7 734 5.4 3.8 5.4
1056 5.7 4.0 5.7 411 6.0 4.2 6.0 387 7.4 5.2 7.4
329B 6.6 4.6 6.6 1390A 6.2 4.4 6.2
1703 5.7 4.0 5.7 M 4.5 3.2 4.5
184A 4.2 2.9 4.2 1791 4.9 3.4 4.9 -AB 6.0 4.2 6.0 117 6.0 4.2 6.0 46A 5.9 4.1 5.9 856 5.9 4.2 5.9
1725 5.3 3.7 5.3 846A 6.1 4.3 6.1 546A 6.1 4.3 6.1 639A 6.1 4.3 6.1 691B 7.7 5.4 7.7
1324DP 3.3 2.3 3.3
Bench Mark
1002DP 185DP 744 734 1056 411 387 329 1390 1703 M 184A 1791 -AB 117 46A 856 1725 846A 546A 639A 691B 1324DP
43
Table 3.9. Comparison of Subsidence Results.
Maraven Values [mm]
84-86 86-88
+1 -4 +20 +21
-3 -2 +32 -34 -38 -43 +13 -6 -12 0 -42 -34 -14 -9
-6 -14 -6 -9
+2 -4 -10 -10
-115 -88 -10 -4 -14 -4 -21 -36
-8 +2 -58 -48
-5 -13 -9 -18
-22 -9 1 -2
Author Values [mm]
84-86 86-88
-7 -12 +31 +27
+4 -9 +37 -38 -33 -47
+24 -11 -7 -4
-37 -35 -8 -10 -1 -13 -1 -6
-13 0 -6 -6
-111 -86 -7 -2
-11 -2 -19 -34
-4 +3 -54 -48
-2 -11 -7 -17
-21 -8 0 -2
Discrepancies (Maraven-Author)
A[mm] A[mm] 84-86 86-88
+8 +8 -11 -6
-7 +7 -5 +4 -5 +4
-11 +5 -5 +4 -5 +1 -6 +1 -5 -1 -5 -3
+15 -4 -4 -4 -4 -2 -3 -2 -2 -2 -2 -2 -4 -1 -4 0 -3 -2 -2 -1 -1 -1
+1 0
44
Table 3.1 0. Summary of Standard Deviations for the Author Computed Subsidence Values.
Bench cr[mm] cr[mm] cr[mm] Mark 84-86 84-88 86-88
1002DP 5.5 6.4 5.5 185DP 5.1 5.9 5.1
744 5.7 6.6 5.7 734 6.6 7.6 6.6
1056 6.9 8.1 6.9 411 7.3 8.5 7.3 387 9.0 10.5 9.0
329B 8.0 9.3 8.0 1390A 7.6 8.8 7.6
1703 7.0 8.1 7.0 M 5.5 6.4 5.5
184A 5.1 5.9 5.1 1791 6.0 6.9 6.0 -AB 7.3 8.5 7.3 117 7.3 8.5 7.3 46A 7.2 8.3 7.2 856 7.2 8.3 7.2
1725 6.5 7.5 6.5 846A 7.5 8.6 7.5 546A 7.5 8.6 7.5 639A 7.5 8.6 7.5 691B 9.4 10.9 9.4
1324DP 4.0 4.7 4.0
45
(see Table 3.10). Table 3.9 shows again the influence of the aforementioned systematic effect in the order of -4 mm between the 1984-1986 campaigns when the author's
computations are compared with the Maraven data.
For the 1986-1988 campaigns the effect varies from +8 mm near bench mark 1329DP
to about -2 mm on the points near Bachaquero. This may be due to the positive shift
introduced by the Maraven method in the fixed point 1329 DP (Table 3.2) for the 1988 campaign computation. In conclusion, the total uncertainty in the Maraven calculated
subsidence estimates reaches 20 to 30 mm at the 95% confidence level.
3.6.7 Final accuracy evaluation of the subsidence using the UNB Generalized
Method
One further step into the analysis of the subsidence computation arises from the
application of the UNB Generalized Method through the least squares fitting of a selected
deformation model to the observed displacements. In Section 3.6.4, the weighted similarity
transformation was used to determine the "best" displacements .out of the original datum
dependent displacements estimated from two separate static adjustments showing only the reference BM's. For further analysis, the estimated "best" displacements between the 1984
and 1988 campaigns are listed in Table 3.11. On the basis of the observed displacements
and their associated confidence levels, single point displacements and a stable block of
reference points could be identified.
As discussed in Chapter 2, points showing significant movements could be modelled
as separate individual blocks and stable points (i.e. points that do not show significant movement) could be modelled together as a stable block. Once the deformation trend is identified the original displacements together with their variance-covariance matrix are used
in the model fitting process.
Rigid body displacement models similar to equation (2.7) may be written as:
wj (x, y) = 0 and wk(x, y) = ak
46
where j represents the block of all stable points and k represents non-stable points treated as separate rigid blocks with individual rigid body displacement ak with respect to the stable
block. Thus the general model could be expressed as
d + o =Be where d is the vector of subsidence values estimated from the minimally constrained
adjustments of both epochs; c is the vector of unknown parameters;
B is the design matrix of the deformation model formed by rows of zeroes for the
stable points and unit elements in the columns corresponding to the parameters of the
unstable points; and
o is the vector of residuals.
A comprehensive explanation of the estimation of c may be encountered in Chen et al.
[1988]. The estimated parameters between the 1988 and 1984 campaigns together with their
corresponding standard deviations and significance levels for some selected BM's are
shown in Table 3.12. The global test to verify the appropriateness of the above model at a
0.95 confidence level passes, i.e., the inequality cr2of
47
accuracies would then be significantly improved. A maximum standard deviation error of
5.5 mm is shown in Table 3.12. The application of this methodology will also remove most
systematic errors arising from the aforementioned shifts.
Table 3.11. "Best" Weighted Displacements (88-84)
BM 84-88 Significance BM 84-88 significance [mm] level [mm] level
1175DP 14.8 0.88 184A 25.7 > 0.99 1329DP 6.4 0.40 1791 0.0 0.0 1326DP 11.2 0.64 -AB -185.5 > 0.99 1002DP 4.1 0.34 117 2.1 0.21 185DP 69.7 > 0.99 46A -0.9 0.09 744 4.1 0.35 856 -41.5 > 0.99 734 8.7 0.68 1725 10.3 0.73 1056 -69.1 > 0.99 846A -90.3 > 0.99 411 18.0 0.94 546A -1.4 > 0.13 387 -0.4 0.03 580A* -769.3 > 0.99 329B -61.6 > 0.99 639A -13.3 0.79 l390A -7.2 0.64 691B -16.9 0.83 1703 -3.0 0.34 1324DP 9.4 0.60 M 5.i 0.87
*reconstructed in 1986
48
Table 3.12. Estimated Model Parameters (88-84).
Bench ak O'a Significance Remarks Mark (mm) k Level
(mm)
185DP 62.6 3.3 > 0.99 Global test on 1056DP -74.6 3.9 > 0.99 Deformation 411 10.5 3.7 .99 Model passes 329B -58.6 4.8 > 0.99 184A -18.9 3.1 > 0.99 1.08 < 2.17 -AB -183.5 5.5 > 0.99 (F 17 ,20;0.95) 856 -42.2 5.3 > 0.99 846A -87.7 5.0 > 0.99 580A* -757.0 4.9 > 0.99
* re-constructed in 1986
4. ACCURACY OF GPS DERIVED HEIGHT DIFFERENCES
There are two basic factors affecting the accuracy of GPS observations. These are
the range error and the geometry of the satellites. According to Mertikas et al. [ 1986], the range error is expressed by the User Equivalent Range Error (UERE) and the geometry by the Geometric Dilution Of Precision (GDOP), which for the vertical direction is referred to as VDOP. The UERE represents the overall effect of all the observational errors arising
from orbit uncertainties, signal propagation errors and receiver related errors. The effect of
the UERE ( O'p) and the VDOP may be combined to yield the total error in a derived height ( crh) through the simple relationship
crh = VDOP O'p . (4.1) Thus, in order to analyse the accuracy of the GPS derived height difference observations, a
brief discussion of the geometry and the most significant observational errors is given in this
chapter. Results from three test GPS campaigns performed in Venezuela between April
1987 and April1988 are also presented. Finally, brief discussions on accuracy expectations
and cost evaluation for the future Costa Bolivar GPS campaigns are also included.
Differential GPS positioning and successful ambiguity resolution has been assumed
throughout the discussion.
4.1 Satellite Geometry
The effect of satellite geometry is generally represented by the GDOP which is a
scalar measure of the overall geometrical strength of an immediate point positioning
solution. Although the main concern herein is relative positioning, and since the baselines
are short, the average GDOP within the observation period has been assumed to give still a
valid measure of the geometrical strength of the solution and will be used under this context
50
throughout the thesis. Wells et al. [1986] point out that for long base lines in the order of
thousands of kilometres this does not hold. The GDOP is computed from the square root of
the trace of the cofactor matrix obtained in a position fix using pseudoranges to at least four
observed satellites. This is equivalent to a distance resection ~lution with unit weights.
Thus,
(4.2)
where q2cp, q2N q2h and q21 represent the co-factors of the latitude, longitude, height and
time coordinates respectively, obtained from the cofactor matrix of the estimated position
parameters. The value of the GDOP varies with time and user location since it depends on
the movement of the satellites and satellite coverage.
The selection of different components in ( 4.2) leads to other geometrical scalars such as PDOP, HDOP, or VDOP, for three dimensional, horizontal and vertical positioning
respectively. For instance, the VDOP is obtained from
VDOP=qh . (4.3)
Up to the present time (January 1989) with the available prototype constellation, the geometry of the satellites has been rather poor in some parts of the world. VDOP values in
the order of 4.5 to 5.0, which for high accuracy requirements may be considered as large,
were common in the Venezuelan GPS test campaigns to be discussed later.
For the future 24 satellite constellation, significant improvements in VDOP values are
expected, especially near the equator where the satellite distribution will be the most
uniform. Santerre [ 1988] shows that the best satellite coverage will be obtained at low
latitudes. VDOP values smaller than 3 may be expected [Milliken and Zoller, 1980].
4. 2 Observational Errors
4.2.1 Orbit related errors
The orbit related errors are induced by inaccuracies in the measured or predicted orbit
51
of the satellites. The uncertainty in the broadcast ephemeris is considered to be in the order
of 20 to 25 m and for the precise ephemeris, as provided by the U.S. Department of
Defense (DoD), in the order of 5 to 10m [Beutler et al., 1986]. The effect is reduced by relative positioning and can be hnproved by more accurate
ephemeris models among other techniques. There are three general uncertainties related to
the orbit -- the radial, the out of plane, and the along track biases. According to Beutler et
al. [1987b], the along track biases affect more significantly-the height components than the
radial biases. The error introduced in the height difference (eM) is said to be equal to L\s
el1h =cos (Az8 - Az\J- b , (4.4) p
where L\s is the magnitude of the along track bias,
Azb is the azimuth of the baseline,
Azs is the azimuth of the orbital plane of a particular satellite being tracked,
p is the range to the satellite, and
b is the length of the baseline.
The maximum error is expected when the baseline orientation coincides with the
orbital plane orientation. The error is proportional to length of the baseline. A maximum
scale error of 1 to 2 ppm is normally expected when using the broadcast ephemerides. For
precise work, the use of precise ephemerides will be more appropriate.
4.2.2 Tropospheric effect
The tropospheric effect is probably the major limitation of GPS in deformation monitoring applications, especially in vertical deformation studies. The effect consists of a
delay in the satellite signal as it propagates through the innermost 80 km of the atmosphere.
The total refractive effect of the troposphere can be separated into two main effects - the
effect of the dry and the wet components. The dry component is responsible for 90% of the
total refractivity and can be modelled from surface meterological data with an accuracy of
52
about 1% [Hopfield, 1969], which is equivalent to a range error of about 1 to 6 em. The
wet component is responsible for the remaining 10% but may be modelled to about 10% to
20%, due to the variable water vapor distribution in space and time [Lachapelle et al. 1988].
The effect is reduced-in relative positioning particularly for short baselines, provided similar
conditions prevail at both ends of the line. According to Beutler et al. [1987b], the effect
has a large influence on the height component with an amplification factor of 1/cos z (where z is the maximum zenith angle to the satellite) with respect to the range bias, the effect becoming larger at low elevation angles. Using a simulated continuous satellite distribution,
Geiger [1988] estimated an average amplification factor of 3. Most of the problems encountered with tropospheric modelling are due to
inaccuracies in the standard meteorological equipment and local microclimate effects which
do not reflect the upper atmospheric conditions at each station. As a result, biased
corrections may be expected. To illustrate this further, a table of zenith range errors arising
from errors in metereological data using Hopfield's model has been taken from
Chrzanowski et al. [1988] and is shown in Table 4.1.
Table 4.1. The zenith range error due to errors in meteorological data (taken from Chrzanowski et al. 1988).
Temperature (T("C)
0 15 30
0 15 30
Pressure p(mb)
.1000 1000 1000 1000 1000 1000
Humidity H(%)
50 50 50
100 100 100
dp/dp [mrn/mb]
2.3 2.3 2.3 2.3 2.3 2.3
dp/dT [mrn/"C]
2.3 5.0 9.8 4.5 9.9
19.6
dp/dH [mrn/%]
0.8 1.7 3.8 0.8 1.7 3.8
53
Notice the large range errors introduced by the errors in temperature and humidity
under very hot and humid conditions. This is an indication of the sensitivity of the solution
to the tropospheric effect in tropical climates.
To reduce this effect in small networks, it is common not to use the surface
metereological data at each receiver site directly but to average the data and use a local
atmosphere model for each session or for the whole campaign. This may be valid for
networks with small height differences but for mountainous areas it may yield biased results
[Gurtner et al., 1987].
In precise applications the use of balloon or helicopter data collected above each GPS
site may be utilized. Pedroza [1988] gives details on tests conducted with this technique
during the third GPS campaign at the Costa Bolivar oil fields using instrumentation designed
and constructed at the University of New Brunswick. To determine the water vapour
pressure in the signal path water vapour radiometry (WVR) has been used, but presently the instrumentation is very expensive and difficult to handle and calibrate [Lachapelle et al.,
1988]. Another alternative to tropospheric modelling is to estimate a tropospheric scale
factor directly in the adjustment [Santerre, 1988] but it appears to be highly correlated with the height component.
More research on the influence of the troposphere is needed especially at low latitudes
where the tropospheric effect is more critical due to the high relative humidity and high
temperatures usually encountered.
4.2.3 Ionospheric effect
The ionospheric effect consists of the propagation delay of the satellite signals due to
interaction with the charged ions present in the upper atmosphere (from about 80 to 1000 km altitude). The relative effect (~e) corresponds to a scale error which is a function of the total electron content (TEC), frequency of the carrier and base line length (b). Beutler et al. [1987b] present a formula to quantify the effect on the Ll carrier as:
54
&! -17 b= -0.7 X 10 TEC. (4.5) The TEC depends directly on solar flux. Thus, it varies with respect to time of the
year, time of the day, latitude and direction to the satellite. The highest TEC distributions
are found near the equator and in the Auroral regions. Beutler et al. [1987b] estimated
values with the L1 carrier varying between 0.35 ppm at night to about 3.5 ppm in the early
local afternoon hours for a user location at 20" latitude. The effect is corrected accurately
using dual frequency observations but a remainder may be left in areas or at times of high
TEC [Wells, et al., 1986]. The dual frequency correction increases the noise level of the
observations by a factor of 3.3 [Kleusberg, 1986]'~ In single frequency observations
empirical formulae may be used but the level of accuracy of the correction is rather poor
(50% - 75% ). When the baselines are short the effect is expected to cancel by the differencing, if one assumes that the signal propagates through a homogeneous ionospheric
layer. However, this may not generally be the case since local irregularities may affect the
signals to each receiver differently. Thus, the integer nature of the differenced ambiguities is
corrupted and the data processing becomes difficult.
Another critical effect of ionospheric irregularities, specifically the so-called
ionospheric scintillations [Lachapelle et al. 1988] is the fading of the signals. This may
cause loss of phase lock and a large number of cycle slips in the data. Lachapelle et al.
[1988] point out that multiplexing and sequential channel receivers may be more affected
than multichannel receivers due to the less favourable signal to noise ratio in these receivers.
These irregulariti